Assume that the situation can be expressed as a linear cost function. Find the cost function. Fixed cost is $300; 50 items cost $800 to produce. The linear cost function is C(x) =
Assume that the situation can be expressed as a linear cost function. Find the cost function. Fixed cost is $300; 50 items cost $800 to produce. The linear cost function is C(x) =
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
Chapter2: Graphical And Tabular Analysis
Section2.2: Graphs
Problem 15E
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![### Understanding Linear Cost Functions
In this section, we will explore how to determine the cost function for a given situation using linear equations. A cost function typically represents the total cost \( C(x) \) to produce \( x \) items.
---
**Example Problem:**
**Assume that the situation can be expressed as a linear cost function. Find the cost function.**
- Fixed cost is $300
- The cost to produce 50 items is $800
---
**Solution:**
1. **Identify the components of the cost function:**
- **Fixed Cost:** This is the cost that does not change with the number of items produced. For this problem, the fixed cost is provided as $300.
- **Variable Cost:** This is the cost that varies with the number of items produced. Our goal is to determine the variable cost per item.
2. **Determine the total cost equation:**
Given that the cost function is linear, it can be expressed in the form:
\[
C(x) = Fixed\ Cost + (Variable\ Cost \times Number\ of\ Items)
\]
where \( C(x) \) is the total cost to produce \( x \) items.
3. **Using the given data to find the variable cost:**
We know the fixed cost (\( FC \)) is $300. We also know that producing 50 items costs $800 in total. Thus:
\[
C(50) = 800
\]
Using the cost function form:
\[
800 = 300 + (Variable\ Cost \times 50)
\]
Simplifying, we find that the variable cost per item (VC) is:
\[
800 = 300 + 50 \times VC
\]
\[
800 - 300 = 50 \times VC
\]
\[
500 = 50 \times VC
\]
\[
VC = 10
\]
Hence, the variable cost per item is $10.
4. **Express the linear cost function:**
Substituting the values obtained into the cost function form:
\[
C(x) = 300 + 10x
\]
---
**Conclusion:**
Therefore, the linear cost function is:
\[
C(x) = 300 + 10x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F11a64bba-74c0-4fd7-b16b-3d8fff0a6ac2%2F7a1309b9-c528-45b2-be7c-2c5ed2807a68%2Fuv2e94x_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Linear Cost Functions
In this section, we will explore how to determine the cost function for a given situation using linear equations. A cost function typically represents the total cost \( C(x) \) to produce \( x \) items.
---
**Example Problem:**
**Assume that the situation can be expressed as a linear cost function. Find the cost function.**
- Fixed cost is $300
- The cost to produce 50 items is $800
---
**Solution:**
1. **Identify the components of the cost function:**
- **Fixed Cost:** This is the cost that does not change with the number of items produced. For this problem, the fixed cost is provided as $300.
- **Variable Cost:** This is the cost that varies with the number of items produced. Our goal is to determine the variable cost per item.
2. **Determine the total cost equation:**
Given that the cost function is linear, it can be expressed in the form:
\[
C(x) = Fixed\ Cost + (Variable\ Cost \times Number\ of\ Items)
\]
where \( C(x) \) is the total cost to produce \( x \) items.
3. **Using the given data to find the variable cost:**
We know the fixed cost (\( FC \)) is $300. We also know that producing 50 items costs $800 in total. Thus:
\[
C(50) = 800
\]
Using the cost function form:
\[
800 = 300 + (Variable\ Cost \times 50)
\]
Simplifying, we find that the variable cost per item (VC) is:
\[
800 = 300 + 50 \times VC
\]
\[
800 - 300 = 50 \times VC
\]
\[
500 = 50 \times VC
\]
\[
VC = 10
\]
Hence, the variable cost per item is $10.
4. **Express the linear cost function:**
Substituting the values obtained into the cost function form:
\[
C(x) = 300 + 10x
\]
---
**Conclusion:**
Therefore, the linear cost function is:
\[
C(x) = 300 + 10x
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